PhD Thesis Proposals

Stability and bifurcation of piecewise-homogeneous closed elastic rods

Tubular epithelial ducts motivate a model of closed elastic rods with a jump in natural curvature at an interface. Existing work identifies several equilibrium families and the morphological separatrices between them; the open problem is to determine their local energetic stability and complete the two-parameter bifurcation diagram in the interface position $s_0$ and curvature jump $\kappa$. The project combines constrained variational analysis, the spectrum of the constrained Hessian, numerical continuation, and singular asymptotics.

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Phase-field modeling of surface growth coupled with diffusion and its stability analysis

Motivation

Surface growth, i.e., the accretion of a solid onto a surface, occurs in several contexts of physical, technological, and biological interest. Conventional areas of application are:

In all these cases, the formation of a new material surface is strongly dependent on chemo-mechanically coupled processes that occur within the material, which may involve thermal or diffusion effects.

Surface growth may induce mechanical instability, which plays a crucial role in morphogenesis. Most descriptions of growth-induced instability are concerned with a prescribed growth path.

Question

Suppose that we are working with a diffuse interface model. Then the question is: what is the extension of the body?

We write the stress work as:

\[\int_{\text{Space}} \varphi N \delta \varepsilon\]

or as:

\[\int_{\text{Space}} N \delta \varepsilon\]

The constitutive equation would be:

\[N = \mathbb{C}(c) \varepsilon\]

where $c$ is the concentration of solvent.

References